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Two Approximation Results for Divergence Free Measures

Jesse Goodman, Felipe Hernandez, Daniel Spector

Abstract

In this paper we prove two approximation results for divergence free measures. The first is a form of an assertion of J. Bourgain and H. Brezis concerning the approximation of solenoidal charges in the strict topology: Given $F \in M_b(\mathbb{R}^d;\mathbb{R}^d)$ such that $\operatorname*{div} F=0$ in the sense of distributions, there exist oriented $C^1$ loops $Γ_{i,l}$ with associated measures $μ_{Γ_{i,l}}$ such that \[ F= \lim_{l \to \infty} \frac{\|F\|_{M_b(\mathbb{R}^d;\mathbb{R}^d)}}{n_l \cdot l} \sum_{i=1}^{n_l} μ_{Γ_{i,l}} \] weakly-star in the sense of measures and \[ \lim_{l \to \infty} \frac{1}{n_l \cdot l} \sum_{i=1}^{n_l} \|μ_{Γ_{i,l}}\|_{M_b(\mathbb{R}^d;\mathbb{R}^d)} = 1. \] The second, which is an almost immediate consequence of the first, is that smooth compactly supported functions are dense in \[ \left\{ F \in M_b(\mathbb{R}^d;\mathbb{R}^d): \operatorname*{div}F=0 \right\} \] with respect to the strict topology.

Two Approximation Results for Divergence Free Measures

Abstract

In this paper we prove two approximation results for divergence free measures. The first is a form of an assertion of J. Bourgain and H. Brezis concerning the approximation of solenoidal charges in the strict topology: Given such that in the sense of distributions, there exist oriented loops with associated measures such that weakly-star in the sense of measures and The second, which is an almost immediate consequence of the first, is that smooth compactly supported functions are dense in with respect to the strict topology.

Paper Structure

This paper contains 9 sections, 6 theorems, 78 equations, 1 figure.

Table of Contents

  1. Main Results and Discussion
  2. Main Results
  3. Discussion
  4. Smirnov's Theorem
  5. Dimension of Singularities of Measures with Differential Constraints
  6. From Curves to Divergence Free Measures: Estimates for Integrals Operators
  7. Uniformity over Curves
  8. Approximating general integrals by sums
  9. Proofs

Key Result

Theorem 1.1

Suppose $F \in M_b(\mathbb{R}^d;\mathbb{R}^d)$ is such that $\operatorname*{div} F=0$ in the sense of distributions. Then there exist oriented $C^1$ closed curves $\Gamma_{i,l}$ with associated measures $\mu_{\Gamma_{i,l}}$ such that weakly-star in the sense of measures and

Figures (1)

  • Figure 1: A depiction of the smoothing of corners.

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Theorem 1.1 in HS
  • Lemma 1.4
  • Theorem 2.1
  • proof : Proof of \ref{['intOverClBySums']}
  • Theorem 3.1
  • proof : Proof of \ref{['lln-cor']} using \ref{['intOverClBySums']}
  • proof : Proof of \ref{['bbassertion']}
  • proof : Proof of \ref{['density']}